# Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm - PowerPoint PPT Presentation

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm. Lecture XXVI. Change in the Marginal Cost. Shares of Marginal Cost Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price.

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Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

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## Differential Models of Production: Change in the Marginal Cost and the Multi-Product Firm

Lecture XXVI

### Change in the Marginal Cost

• Shares of Marginal Cost

• Since both total and marginal cost depend on output levels and input prices, we start by considering marginal share of each input price

• Based on this definition, we define a Firsch price index for inputs as

• Completing the single output model

### Multiproduct Firm

• Expanding the production function to a multiproduct technology

• Expanding the preceding proof

• Computing the first-order conditions

• Now we replicate some of the steps from the preceding lecture, allowing for multiple outputs.

• Taking the differential of the first-order condition with respect to each output

• Again note by the first-order condition

• Thus

• With

• Differentiating with respect to the input prices yields the same result as before

• Slightly changing the preceding derivation by differentiating the production function by a vector of output levels, holding prices and other outputs constant yields

• Multiplying through by γyields

• Using the tired first-order conditions

• With

• Differentiating the production function with respect to yields

• Collecting these equations:

• Differentiating the first-order conditions with respect to ln(z’)

• Differentiating the first-order conditions with respect to ln(p’)

• Differentiating the production function with respect to ln(z’)

• Differentiating the production function with respect to ln(p’)

• The extended form of the differential supply system is then.

• Starting with the total derivative of ln(q)

• Premultiplying by F

• Note by the results from Barten’s fundamental matrix

• θir is the share of the ith input in the marginal cost of the rth product.

• Summing this marginal cost over all inputs

• Defining the matrix

### Introduction of Quasi-Fixed Variables

• Expanding the differential model further, we introduce quasi-fixed variables into the production set

• Following Livanis and Moss, the differential supply function for this specification becomes

• Starting with the input demand system, we add a random disturbance relying on the theory of rational random behavior (RRB, Theil 1975):